SOLVE A SYSTEM OF TWO LINEAR EQUATIONS IN TWO VARIABLES BY GRAPHING. SOLVE A SYSTEM OF TWO LINEAR...
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Transcript of SOLVE A SYSTEM OF TWO LINEAR EQUATIONS IN TWO VARIABLES BY GRAPHING. SOLVE A SYSTEM OF TWO LINEAR...
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Solve a system of two linear equations in two variables by graphing. Solve a system of two linear equations in two variables using the substitution and the
elimination methods. Use systems of two linear equations to solve applied
problems.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
9.1 Systems of Equations in Two Variables
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Systems of Equations
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
A system of equations is composed of two or more equations considered simultaneously.
Example: 5x y = 5
4x y = 3
This is a system of two linear equations in two variables. The solution set of this system consists of all ordered pairs that make both equations true. The ordered pair (2, 5) is a solution of this system.
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Solving Systems of Equations Graphically
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
When we graph a system of linear equations, each point at which the graphs intersect is a solution of both equations and therefore a solution of the system of equations.
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Solving Systems of Equations Graphically
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Let’s solve the previous system graphically.
5x y = 5
4x y = 3
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Types of Solutions
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Graphs of linear equations may be related to each other in one of three ways.
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Practice
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Method 1 – Substitution
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
The substitution method is a technique that gives accurate results when solving systems of equations. It is most often used when a variable is alone on one side of an equation or when it is easy to solve for a variable. One equation is used to express one variable in terms of the other, then it is substituted in the other equation.
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Example
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Use substitution to solve the system
5x y = 5,
4x y = 3.
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Method 2 – Elimination
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Using the elimination method, we eliminate one variable by adding the two equations. If the coefficients of a variable are opposites, that variable can be eliminated by simply adding the original equations. If the coefficients are not opposites, it is necessary to multiply one or both equations by suitable constants, before we add.
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Example
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Solve the system using the elimination method.
6x + 2y = 4
10x + 7y = 8
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Another Example
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Solve the system.
x 3y = 9
2x 6y = 3
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Another Example
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Solve the system. 9x + 6y = 48 3x + 2y = 16
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SubstitutionSubstitution EliminationElimination
Practice
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Application--Example
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Ethan and Ian are twins. They have decided to save all of the money they earn, at their part-time jobs, to buy a car to share at college. One week, Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256. The next week, Ethan worked 12 hours and Ian worked 16 hours and they earned $324. How much does each twin make per hour?
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Example
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At Max’s Munchies, caramel corn worth $2.50 per pound is mixed with honey roasted mixed nuts worth $7.50 per pound in order to get 20 lb. of a mixture worth $4.50 per pound. How many of each snack is used?
Carmel corn
Nuts Mixture
Price per pound
$2.50 $7.50 $4.50
Number of pounds
x y 20
Value of Mixture
2.50x 7.50y 4.50(20) = 90
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Example
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An airplane flies the 3000-mi distance from Los Angeles to New York, with a tailwind, in 5 hr. The return trip, against the wind, takes 6 hr. Find the speed of the airplane and the speed of the wind. Distance Rate Time
With Tailwind 3000 p + w 5With headwind 3000 p – w 6